Problem: Mrs. Wheeler prepares a list of $43$ US presidents, $31$ of whom served in the military. Then $8$ students each select a president at random (there can be repeats) for their civics presentations. What is the probability that at least one of the students will select a president who did not serve in the military? Round your answer to the nearest hundredth. $P(\text{at least one not in military})=$
Solution: Strategy In this situation it is much easier to calculate the probability of the event we are looking for (at least one president who did not serve in the military) by calculating the probability of its complement (all presidents who served in the military), and subtracting from $1$. In other words, we can use this strategy: $P(\text{at least one not in military})=1-P(\text{all 8 in military})$ Calculations $\begin{aligned} &\phantom{=}P(\text{at least one not in military}) \\\\ &=1-P(\text{all 8 in military}) \\ \\ &=1-\left(\dfrac{31}{43}\right)^{8} \\ \\ &\approx 1-0.073 \\ \\ &\approx 0.927\end{aligned}$ Answer $P(\text{at least one not in military}) \approx 0.93$